Hindawi Publishing Corporation Boundary Value Problems Volume 2010, Article ID 516260, 13 pages doi:10.1155/2010/516260 Research Article A New Conservative Difference Scheme for the General Rosenau-RLW Equation Jin-Ming Zuo,1 Yao-Ming Zhang,1 Tian-De Zhang,2 and Feng Chang2 School of Science, Shandong University of Technology, Zibo 255049, China School of Mathematics, Shandong University, Jinan 250100, China Correspondence should be addressed to Jin-Ming Zuo, zuojinming@sdut.edu.cn Received 28 May 2010; Accepted 14 October 2010 Academic Editor: Colin Rogers Copyright q 2010 Jin-Ming Zuo et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited A new conservative finite difference scheme is presented for an initial-boundary value problem of the general Rosenau-RLW equation Existence of its difference solutions are proved by Brouwer fixed point theorem It is proved by the discrete energy method that the scheme is uniquely solvable, unconditionally stable, and second-order convergent Numerical examples show the efficiency of the scheme Introduction In this paper, we consider the following initial-boundary value problem of the general Rosenau-RLW equation: ut − uxxt uxxxxt ux up x xl < x < xr , < t < T , 1.1 with an initial condition u x, u0 x xl ≤ x ≤ xr , uxx xl , t uxx xr , t 1.2 and boundary conditions u xl , t u xr , t 0, 0≤t≤T , 1.3 Boundary Value Problems where p ≥ is a integer and u0 x is a known smooth function When p 2, 1.1 is called as usual Rosenau-RLW equation When p 3, 1.1 is called as modified Rosenau-RLW MRosenau-RLW equation The initial boundary value problem 1.1 – 1.3 possesses the following conservative quantities: Q t E t u L2 ux xr u x, t dx xl L2 uxx 2 L2 xr u0 x, t dx Q 0, 1.4 xl u0 L2 L2 u0x u0xx L2 E 1.5 It is known the conservative scheme is better than the nonconservative ones Zhang et al point out that the nonconservative scheme may easily show nonlinear blow up In Li and Vu-Quoc said “ in some areas, the ability to preserve some invariant properties of the original differential equation is a criterion to judge the success of a numerical simulation” In 3–11 , some conservative finite difference schemes were used for a system of the generalized nonlinear Schrodinger equations, Regularized long wave RLW equations, ă Sine-Gordon equation, Klein-Gordon equation, Zakharov equations, Rosenau equation, respectively Numerical results of all the schemes are very good Hence, we propose a new conservative difference scheme for the general Rosenau-RLW equation, which simulates conservative laws 1.4 and 1.5 at the same time The outline of the paper is as follows In Section 2, a nonlinear difference scheme is proposed and corresponding convergence and stability of the scheme are proved In Section 3, some numerical experiments are shown A Nonlinear-Implicit Conservative Scheme In this section, we propose a nonlinear-implicit conservative scheme for the initial-boundary value problem 1.1 – 1.3 and give its numerical analysis 2.1 The Nonlinear-Implicit Scheme and Its Conservative Law For convenience, we introduce the following notations xj xr jh, tn nτ, j 0, 1, , J, n 0, 1, , T τ N, 2.1 where h xr − xl /J and τ denote the spatial and temporal mesh sizes, un ≡ u xj , tn , j n Uj ≈ u xj , tn , respectively, n Uj n Uj x n Uj n − Uj τ t n Uj x n Uj Un n Uj , x , n Uj n n Uj − Uj x 1/2 Un , Un , h n Uj Un n Uj , n n Uj − Uj−1 h x , J−1 n Uj , Un , V n n Uj Vjn , h j ∞ n max Uj , 1≤j≤J 2.2 Boundary Value Problems and in the paper, C denotes a general positive constant, which may have different values in different occurrences p−1 i p−i 2/ p Since up x x , then the finite difference scheme for the i u u problem 1.1 – 1.3 is written as follows: n Uj t n − Uj xxt n Uj n Uj xxxxt p−1 1/2 x p 1i n Uj n U0 n UJ u0 xj , n U0 0, j n Uj 1/2 0, 1, 2, , J, n UJ xx i 1, 2, , J − 1; j Uj 1/2 xx 0, n p−i 0, x n 1, 2, , N, 2.3 2.4 1, 2, , N 2.5 Lemma 2.1 see 12 For any two mesh functions, U, V ∈ Zh , one has U x, V − U, V x , U x, V − U, V x , U x V, U U, U n Furthermore, if U0 xx n UJ xx − V xx x, − U x, U 2.6 , − Ux x 0, then xx U, U Uxx xxxx 2.7 Theorem 2.2 Suppose that u0 ∈ H0 xl , xr , then scheme 2.3 – 2.5 is conservative in the senses: Qn En n U 2 h J−1 n U 2j j Qn−1 n U x n U xx ··· Q0 , En−1 2.8 ··· E0 2.9 Proof Multiplying 2.3 with h/2, according to boundary condition 2.5 , and then summing up for j from to J − 1, we have h J−1 n Uj 2j 1 n − Uj 2.10 Boundary Value Problems Let h J−1 n U 2j j Qn 2.11 Then 2.8 is gotten from 2.10 Computing the inner product of 2.3 with Un 2.5 and Lemma 2.1, we obtain n U 2 t n U x n U xx t t Un 1/2 x , Un 1/2 , according to boundary condition 1/2 κ Un 1/2 , Un 1/2 , Un 1/2 0, 2.12 where κ Un 1/2 , Un p−1 1/2 p Un 1/2 1/2 i Un 1/2 p−i , x Un 1/2 Un Un 1i 2.13 Un According to κ Un 1/2 , Un 1/2 , Un x p−1 1/2 p − − we have κ Un 1/2 , Un 1/2 , Un 1/2 n U p 1/2 0, Un 1/2 i Un 1/2 p−i x i p−1 1 Un 1/2 i x i p−1 p , Un Un 1/2 i , Un Un , Un 2.14 1/2 p−i 1/2 p−i i 1/2 x , Un 1/2 , It follows from 2.12 that t n U x t n U xx t 2.15 Let En n U 2 n U x n U xx Then 2.9 is gotten from 2.15 This completes the proof of Theorem 2.2 2.16 Boundary Value Problems 2.2 Existence and Prior Estimates of Difference Solution To show the existence of the approximations Un n 1, 2, , N for scheme 2.3 – 2.5 , we introduce the following Brouwer fixed point theorem 13 Lemma 2.3 Let H be a finite-dimensional inner product space, · be the associated norm, and g : H → H be continuous Assume, moreover, that there exist α > 0, for all z ∈ H, z α, and z∗ ≤ α ω z , z > Then, there exists a z∗ ∈ H such that g z∗ {ν Let Zh following νj | ν0 Theorem 2.4 There exists Un νJ ν0 νJ xx 0, j xx 0, 1, , J}, then have the ∈ Zh which satisfies scheme 2.3 – 2.5 Proof (by Brouwer fixed point theorem) It follows from the original problem 1.1 – 1.3 that U0 satisfies scheme 2.3 – 2.5 Assume there exists U1 , U2 , , Un ∈ Zh which satisfy scheme 2.3 – 2.5 , as n ≤ N − 1, now we try to prove that Un ∈ Zh , satisfy scheme 2.3 – 2.5 We define ω on Zh as follows: 2ν − 2Un − 2νxx ω ν 2νxxxx − 2Uxxxx 2Uxx τνx τκ ν, ν , 2.17 p−1 i p−i where κ ν, ν 2/ p x Computing the inner product of 2.17 with ν and i ν ν 0, we obtain considering κ ν, ν , ν and νx , ν ν 2 νx 2 νxx − Un , ν ≥2 ν ω ν ,ν 2 νx 2 νxx − n Uxx − n Ux ν ≥ ν 2 νx − νx Un − − n Ux νxx 2 ν νxx Un Un n n Uxx , ν − Uxxxx , ν 2 n Uxx n Ux 2 2.18 n Uxx n n Hence, for all ν ∈ Zh , ν Un Ux Uxx there exists ω ν , ν ≥ It follows ∗ Let Un 2ν − Un , then it can from Lemma 2.3 that exists ν ∈ Zh which satisfies ω ν∗ n ∈ Zh is the solution of scheme 2.3 – 2.5 This completes the proof of be proved that U Theorem 2.4 Next we will give some priori estimates of difference solutions First the following two lemmas 14 are introduced: Lemma 2.5 discrete Sobolev’s estimate For any discrete function {un | j j finite interval {xl , xr }, there is the inequality un ∞ ≤ ε un x where ε, C ε are two constants independent of {un | j j 0, 1, , J} on the C ε un , 0, 1, , J} and step length h 2.19 Boundary Value Problems Lemma 2.6 discrete Gronwall’s inequality Suppose that the discrete function {wn | n , N} satisfies the inequality wn − wn−1 ≤ Aτwn where A, B and Cn n Bτwn−1 Cn τ, 0, 1, 2.20 0, 1, 2, , N are nonnegative constants Then max |wn | ≤ N w0 1≤n≤N Cl e2 A τ B T , 2.21 l where τ is sufficiently small, such that A B τ ≤ N − /2N, N > Theorem 2.7 Suppose that u0 ∈ H0 xl , xr , then the following inequalities Un ≤ C, n Ux ≤ C, Un ∞ ≤ C, n Uxx ≤ C 2.22 hold Proof It is follows from 2.9 that Un ≤ C, n Ux ≤ C, n Uxx ≤ C 2.23 According to Lemma 2.5, we obtain Un ∞ ≤ C 2.24 This completes the proof of Theorem 2.7 Remark 2.8 Theorem 2.7 implies that scheme 2.3 – 2.5 is unconditionally stable 2.3 Convergence and Uniqueness of Difference Solution First, we consider the convergence of scheme 2.3 – 2.5 We define the truncation error as follows: rjn un j t − un j xxt un j xxxxt un j 1/2 p x j p−1 1i un j 1/2 un j 1, 2, , J − 1; then from Taylor’s expansion, we obtain the following i n 1/2 p−i , x 1, 2, , N, 2.25 Boundary Value Problems Theorem 2.9 Suppose that u0 ∈ H0 xl , xr and u x, t ∈ C5,3 , then the truncation errors of scheme 2.3 – 2.5 satisfy rjn O τ2 h2 , 2.26 as τ → 0, h → Theorem 2.10 Suppose that the conditions of Theorem 2.9 are satisfied, then the solution of scheme 2.3 – 2.5 converges to the solution of problem 1.1 – 1.3 with order O τ h2 in the L∞ norm Proof Subtracting 2.3 from 2.25 letting n un − Uj , j n ej 2.27 we obtain rjn n ej t n − ej n ej xxt n ej xxxxt 1/2 κ un j x Computing the inner product of 2.28 with 2en 2r n , en 1/2 t en n ex t 1/2 un 1/2 , un 1/2 j j κ , un j 1/2 n − κ Uj 1/2 n , Uj n ej −κ 1/2 x n , ej 1/2 , un j p p−1 un j h j 1/2 p−1 j 1/2 n , Uj i i 1/2 un j , en 2.29 n n Uj 1/2 , Uj 1/2 p−1 p−i 1/2 n ej h 1/2 i i−1 1/2 ,e p−1 n 1/2 en i un j 1/2 Uin 1/2 i i−1−r n Uj 1/2 r p−i−1 n ej L∞ p−i ≤ C Then by un j 1/2 1/2 un j 1/2 p−i−1−r n ej x i 1/2 p−i x n Uj 1/2 r n ej 1/2 r i en 1/2 r − ≤C n Uj − x un j 2.28 1/2 i J−1 p n − κ Uj J−1 1/2 1/2 From the conservative property 1.5 , it can be proved by Lemma 2.5 that u Theorem 2.7 we can estimate 2.29 as follows: κ un j 1/2 , we obtain t n exx 1/2 n ex n ex 2.30 Boundary Value Problems According to the following inequality 11 n ex ≤ en n exx n ej 2r n , en 1/2 n ex , 1/2 x n , ej ≤ 1/2 ≤ rn 1 en n exx1 , 0, en 2.31 en Substituting 2.30 – 2.31 into 2.29 , we obtain en t n ex t n exx t ≤ rn C en en n ex n ex n exx n exx1 2.32 Let Bn en 2 n ex n exx , 2.33 then 2.32 can be rewritten as Bn − Bn−1 ≤ Cτ τ 2 h2 Cτ Bn − Bn−1 2.34 Choosing suitable τ which is small enough, we obtain by Lemma 2.6 that Bn ≤ C B0 τ2 h2 2.35 From the discrete initial conditions, we know that e0 is of second-order accuracy, then B0 O τ2 h2 2.36 Then we have en ≤ O τ h2 , n ex ≤ O τ It follows from Lemma 2.5, we have en Theorem 2.10 ∞ h2 , ≤ O τ2 n exx ≤ O τ h2 2.37 h2 This completes the proof of Boundary Value Problems Table 1: The errors of numerical solutions at t h 0.4 0.2 0.1 0.05 0.025 un − Un 5.476 1.385 3.474 8.691 2.059 un − Un 327 × 10−2 256 × 10−2 318 × 10−3 419 × 10−4 064 × 10−4 1.958 4.983 1.252 3.134 7.550 60 with τ un/4 − Un/4 / un − Un ∞ 718 × 10−2 761 × 10−3 185 × 10−3 571 × 10−4 730 × 10−5 0.4 0.2 0.1 0.05 0.025 un − Un un − Un 1.164 674 × 10−1 2.940 136 × 10−2 7.357 052 × 10−3 1.837 759 × 10−3 4.283 535 × 10−4 un/4 − Un/4 3.953 296 3.987 130 3.997 412 4.221 051 Table 2: The errors of numerical solutions at t h h for p 4.251 029 × 10−2 1.080 424 × 10−2 2.708 996 × 10−3 6.772 212 × 10−4 1.596 208 × 10−4 un − Un ∞ 3.930 200 3.980 050 3.994 759 4.151 348 60 with τ un/4 − Un/4 / un − Un ∞ ∞/ h for p un/4 − Un/4 3.961 294 3.996 350 4.003 273 4.290 286 ∞/ un − Un ∞ 3.934 592 3.988 283 4.000 165 4.242 688 Theorem 2.11 Scheme 2.3 – 2.5 is uniquely solvable Un − U n , we obtain Proof Assume that Un and U n both satisfy scheme 2.3 – 2.5 , let W n Wjn t − Wjn κ Wjn xxt xxxxt n 1/2 n 1/2 −κ Uj , Uj Wj0 j n Uj 1/2 − Ujn x Ujn 1/2 , Ujn 1/2 1/2 x 0, 2.38 0, 1, , N Similarly to the proof of Theorem 2.10, we have Wn n Wx n Wxx 2.39 This completes the proof of Theorem 2.11 Remark 2.12 All results above in this paper are correct for initial-boundary value problem of the general Rosenau-RLW equation with finite or infinite boundary Numerical Experiments In order to test the correction of the numerical analysis in this paper, we consider the following initial-boundary value problems of the general Rosenau-RLW equation: ut − uxxt uxxxxt ux up x 0